Delay-dependent stabilization of linear systems with time-varying state and input delays.

*(English)*Zbl 1093.93024Summary: The integral-inequality method is a new way of tackling the delay-dependent stabilization problem for a linear system with time-varying state and input delays:

$$\dot{x}\left(t\right)=Ax\left(t\right)+{A}_{1}x(t-{h}_{1}\left(t\right))+{B}_{1}u\left(t\right)+{B}_{2}u(t-{h}_{2}\left(t\right))\xb7$$

In this paper, a new integral inequality for quadratic terms is first established. Then, it is used to obtain a new state- and input-delay-dependent criterion that ensures the stability of the closed-loop system with a memoryless state feedback controller. Finally, some numerical examples are presented to demonstrate that control systems designed based on the criterion are effective, even though neither $(A,{B}_{1})$ nor $(A+{A}_{1},{B}_{1})$ is stabilizable.

##### MSC:

93D15 | Stabilization of systems by feedback |

93C23 | Systems governed by functional-differential equations |